\chapter{Introduction}
\label{cha:intro}
\section{Spontaneous Pattern Formation}

The spontaneous emergence of regular structure from natural processes has been observed throughout history. Found in nearly every field of science, and even in social and personal interactions, patterns are one of the most recognizable signatures of a nonlinear dynamical system. The study of such systems, and other large classes of phenomena, including chaos and solitons, is the primary goal of the active branch of physics known as \emph{dynamics}. The mathematical tools developed in the field of dynamics have been used successfully to describe a wide range of pattern forming systems in biology, chemistry, computer science, and sociology \cite{Cross_1993aa,Strogatz_2001aa,Lim_2007aa}.

The spontaneous formation of patterns takes place through a variety of mechanisms, each of which has the generic feature of competition between driving forces and dissipation in the system. In Rayleigh-B\'{e}nard convection, a canonical example of pattern formation, patterns are formed in a sheet of fluid contained between two horizontal plates where the driving force is a temperature difference between the top plate and the bottom plate. This driving force overcomes dissipation due to thermal conduction and viscosity, leading to the formation of convection patterns within the fluid.

Another simple example from nonlinear optics is the competition between nonlinearity and dispersion in soliton pulse propagation. In a nonlinear optical medium, pulses with a certain shape experience a nonlinearity  that exactly balances the effects of dispersion, thereby allowing the pulses to propagate without changing shape. In Chapter~\ref{cha:patterns}, I describe in greater detail an example of a pattern-forming system in nonlinear optics.

The quantitative description of pattern formation requires a study of the system dynamics and their stability relative to perturbations. In the general approach, a system is described by a set of differential equations that specify the trajectory of the system's state vector through phase space. Linear stability is determined by applying infinitesimal perturbations to the steady-state solutions of this equation set, and determining if such perturbations grow or shrink as the state of the system evolves. For spatially extended systems, the same concepts apply, although it is often the stability of the Fourier modes of the system that is of interest. Hence, if infinitesimal perturbations applied to a specific mode grow as the system evolves, that mode can give rise to an instability. Instabilities such as this are responsible for pattern formation in systems with two or more dimensions.

The term \emph{pattern selection} refers to the tendency of the system to exhibit patterns with a certain symmetry or orientation. Understanding the pattern selection process is of fundamental importance to understanding the patterns observed in the system because many patterns are allowable solutions to the dynamics equations of the system, yet only a subset of the allowed patterns are typically exhibited. Patterns are selected both by constraints on the system and by the dynamics of the system. The optical patterns that are the subject of this thesis exhibit pattern selection by both mechanisms, although primarily via the system dynamics, in particular through external forcing \cite{Cross_1993aa}.

Given a specific system, and thus specific allowable solutions (\emph{i.e.}, allowed patterns), control of the patterns is limited to choosing from among these solutions. Hence, it is through controlling pattern selection that one can  control the pattern generated by a system. Attempting to control the spontaneous patterns formed by nonlinear processes is not an intrinsically new idea. In fact, attempts to control many aspects of nature are simply attempts to control the patterns that arise from nature's fundamental processes. Weather, ocean currents, tides, and wind are all patterns. There are, however, new applications for controllable pattern-formation, and one such application, controlling the flow of optical information, is the focus of this thesis.

Scientists are generally interested in controlling the flow of information. An early example of this is the diode vacuum tube, where electrons drift from a hot filament (cathode) to a charged plate (anode) allowing current to flow through the device in only one direction. More advanced vacuum tubes use similar principles to amplify electronic signals and perform other information-control functions. Advances in solid state physics led to the development of semiconductor materials and the transistor. Considered one the greatest inventions of the 20th century, the transistor has been responsible for the miniaturization of electronics enabling an enormous range of information-handling devices from cellular telephones to supercomputers. The development of the internet (world-wide-web) was enabled by personal computers and fueled by scientists need for shared information in the high-energy physics community.

To support the growing technological demands of the information age, the field of optics has developed a wide range of devices for generating, transmitting, detecting, and processing optical signals. All-optical devices are capable of operating at much higher bandwidths than electronic devices and more importantly, they are capable of much higher information density. In order to store or process information, an electronic device must displace electrons and separate them with a potential barrier. This process requires power that must ultimately be dissipated as heat. Thermal dissipation places limits on the density and speed of electronic devices; these limits have already been reached by leading-edge technology \cite{Cavin_2006aa}. The development of all-optical devices that are capable of operating with low input power is thus an important step towards improving information technology.

My contribution to this field is to investigate the control of patterns formed in a nonlinear optical system, and the application of such control to the problem of ultra-low-light-level all-optical switching. Nonlinear optical patterns have been observed for nearly 50 years since the invention of the laser and the birth of the field of nonlinear optics \cite{Lugiato_1994aa}. By combining ideas of pattern formation from the nonlinear dynamics community, with nonlinear optics, I have developed a device that demonstrates all-optical switching at ultra-low-light levels.

Prior to this work, many attempts to improve the sensitivity of all-optical switches were based on two assumptions. The first is that all-optical switching cannot be achieved with an input energy density of less than one photon per atomic cross section \cite{Keyes_1970aa}. The second is that, in order to observe all-optical switching, the nonlinear phase shift induced by the input beam must be of the order of $\pi$ radians \cite{Sharping_2002aa,Soljacic_2002aa}. After discussing some instructive examples that serve to introduce and define these concepts, I will describe an all-optical switch that overcomes both of these limitations by exploiting the inherent sensitivity of nonlinear instabilities and the associated pattern-forming processes.

\section{Overview of this Thesis} % (fold)
\label{sub:overview_of_this_thesis}
This thesis describes my investigation of controlling optical patterns generated by nonlinear interactions between laser light and warm rubidium vapor. Controlling these patterns leads to a new type of all-optical switch that can be actuated by less than 600 photons. The sensitivity, measured in units of photons per \lambdasquared, is comparable to the best results from devices based on electromagnetically-induced transparency (EIT). In addition to having high sensitivity, the switch is the first ultra-low-light level device to exhibit transistor-like response. Additionally, the switch output beams are stronger than the input beams, thus making the switch cascadable; one device can be used to drive another, a feature that is required of all practical logic elements. This thesis presents an all-optical switch that meets the requirements for use as an all-optical logic element that operates at ultra-low-light levels. Additionally, I describe the various modifications that have been made to the experimental setup during the course of my research, and the improvements these modifications led to.

Chapter~\ref{cha:opt_switching} presents two simple all-optical switches that illustrate the fundamental concepts behind all-optical switching. Chapter~\ref{cha:patterns} describes theoretically the origin of pattern formation in nonlinear optical systems with counterpropagating beams. Chapter~\ref{cha:patterns_in_Rb} describes my experimental observations of pattern formation in rubidium vapor. I also describe specific aspects of the experimental system that are necessary to generate patterns that are well suited to sensitive all-optical switching. These include beam alignment, vapor cell design, and shielding from external magnetic fields. In Chapter~\ref{cha:switch}, I present a device that operates as an all-optical switch by controlling the orientation of transverse patterns. Results are presented, and discussed in the context of prior work on all-optical switching. Chapter~\ref{cha:nummodel} presents a numerical model for the switch system, and results from this model. With this model, I obtain qualitatively similar results in comparison to the experiment with regards to the time response of the switch and the input power level. Chapter~\ref{cha:toymodel} presents a simple 1D model, based on arguments of symmetry-breaking, that exhibits critical slowing down which can explain the increase in response time for low switch-beam powers. Finally, in Chapter~\ref{cha:conclusion}, I summarize my main results and describe possible future directions for my research.

In greater detail, I begin the discussion of all-optical switching in Chapter~\ref{cha:opt_switching} with two pedagogical examples of all-optical switches. The first switch, shown in Fig.~\ref{fig:intro_phase_switch}, is based on the intensity-dependent refractive index of nonlinear optical media. In such a switch, the nonlinear phase shift experienced by a beam in traversing a nonlinear optical medium can cause constructive or destructive interference when the medium is used in only one arm of an interferometer.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/phase_switch.pdf}
  \end{center}
  \caption[An example all-optical switch based on the nonlinear phase shift]{The intensity-dependent phase shift experienced by a beam propagating through a nonlinear medium can be used to construct an all-optical switch by placing a nonlinear medium in one arm of an interferometer. A strong control beam affects the nonlinear phase shift and thus the interferometer output.}
  \label{fig:intro_phase_switch}
\end{figure}

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/saturation_switch.pdf}
  \end{center}
  \caption[An example all-optical switch based on saturation]{A strong control beam serves to saturate the absorption of a nonlinear medium. This beam thus controls the transmission of a signal beam propagating through the medium.}
  \label{fig:intro_saturation_switch}
\end{figure}

The second switch, shown in Fig.~\ref{fig:intro_saturation_switch}, is based on a medium with saturable absorption; specifically, an ensemble of two-level atoms. With a medium that exhibits saturable absorption, one beam of light can serve to saturate the atomic response of the sample, thus allowing a signal beam to pass through with little absorption. When the control beam is turned off, the medium is no longer being saturated, and the signal experiences strong absorption.

I also review recent work in the field of low-light all-optical switching, and describe several different schemes that have been demonstrated. To facilitate comparison between vastly different approaches, I introduce a metric where the input energy density of an all-optical switch is measured in photons per \lambdasquared. This metric reports the number of photons required to actuate an all-optical switch that has a transverse dimension equal to \lambdasquared, which corresponds, up to a constant, to both the diffraction limit of the interacting fields and the maximum cross-section for a two-level atom. Using this metric, the most sensitive all-optical switch reported to date can be actuated with as few as $10^{-5}$~photons/\lambdasquared~\cite{Zhang_2007aa}.

In Chapter~\ref{cha:patterns}, I describe a simple model system: two optical beams counterpropagating in a medium with Kerr-type nonlinearity. This model assumes that the optical medium has an intensity-dependent index of refraction: $n(\omega)=n_0+n_2 I$, where $n_0$ is the linear refractive index, $n_2$ is the nonlinear refractive index, and $I$ is the intensity of the light. By making this assumption, it can be shown that counterpropagating beams couple through this intensity dependence and are unstable to the growth of off-axis beams. A linear stability analysis is performed on the steady-state solutions to the set of counterpropagating beam equations. This analysis provides a conceptual foundation for the generation of off-axis patterns in counterpropagating beam systems. The instability threshold, shown in Fig.~\ref{fig:intro_threshold_intensity}, is found to be lowest for $K^2 L/2k\simeq3$ where $K$ is the transverse wavevector of the perturbation, $L$ is the medium length, and $k$ is the pump-beam wavevector within the medium. This result implies that, above the instability threshold, the system will spontaneously emit off-axis beams with transverse wavevector $K=\sqrt{6k/L}$. To continue this introduction, I review a qualitative description of the effects of symmetry-breaking on optical pattern formation. Specifically, I discuss the origin of flowerlike patterns and patterns with hexagonal and two-spot symmetry. Finally, I introduce polarization instabilities, and describe several theoretical treatments and their predictions for my experimental system.

\begin{figure}[htbp]
  \centering
    \includegraphics[height=3in]{Figures/threshold_intensity.pdf}
  \caption[Threshold intensity for self-focusing media with phase grating]{The threshold intensity for plane waves counterpropagating within a transparent medium exhibiting an intensity-dependent refractive index, from \cite{Firth_1988aa}.}
  \label{fig:intro_threshold_intensity}
\end{figure}

In Chapter~\ref{cha:patterns_in_Rb}, I present a simple experimental system that gives rise to transverse optical patterns with less than 1~mW of optical pump power. An instability in the system gives rise to mirrorless parametric self-oscillation, which is responsible for generating new beams of light that propagate at an angle to the pump beams and form multi-spot patterns in the far-field, such as those illustrated in Fig.~\ref{fig:intro_hexagons}. I have characterized this instability, and the generated patterns, in terms of several properties of the pump beams: frequency, intensity, size, and alignment. A forward pump beam with 415~$\mu$W of power and a backward pump beam with 145~$\mu$W of power, both detuned to the high-frequency side of the $^{87}$Rb $F=1 \rightarrow F'=1$, $D_2$ resonance, generate $\sim$3~$\mu$W of optical power. 

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/example_pattern.pdf}
  \end{center}
  \caption[Example pattern formed by a counterpropagating beam system]{Beams counterpropagating through a nonlinear medium give rise to transverse structure in the field in the plane perpendicular to the direction of propagation. a) Beams and nonlinear medium. b) Hexagonal pattern. c) Pump beam transmitted off-resonance (for reference)}
  \label{fig:intro_hexagons}
\end{figure}

This instability is also responsible for the formation of optical patterns. The form of these patterns is two or more spots arranged along a ring corresponding to the projection of a cone of light onto the plane of measurement. The angle between the cone and the pump-beam axis is $\theta\simeq4$~mrad. Patterns with hexagonal symmetry are observed in addition to patterns with up to 18 spots. Increasing the pump beam intensity or the pump beam size leads to patterns with finer transverse scales, and in general, to patterns with a larger number of spots. Just above threshold, for all pump beam sizes studied, the pattern consists of a pair of spots symmetrically located across the pump-beam axis from one another. 

Additionally, I observe a secondary modulational instability that gives rise to fluctuations in the intensity of the generated light. The frequency of these fluctuations depends directly on the angle $\theta_p$ between the counterpropagating pump beams. For well-aligned pump beams, this instability is greatly suppressed, for pump beams misaligned by $\theta_p\simeq0.4$~mrad, the instability has a characteristic frequency of $\sim$245~kHz.

In Chapter~\ref{cha:switch}, I demonstrate the application of transverse optical patterns to ultra-low-light all-optical switching. The system described in Chapter~\ref{cha:patterns_in_Rb} generates patterns that are extremely sensitive to perturbations. A perturbation in the form of a weak switch beam injected into the nonlinear medium is suitable for controlling the orientation of the generated patterns and thus operating as a switch where each state of the pattern orientation corresponds to a state of the switch. This controlled pattern rotation is illustrated in Fig.~\ref{fig:intro_cone_switch}.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/dawes_cone_3.jpg}
  \end{center}
  \caption[Transverse optical pattern rotates when a switch beam is injected into rubidium vapor]{A two-spot transverse optical pattern rotates when a switch beam is injected into the rubidium vapor.}
  \label{fig:intro_cone_switch}
\end{figure}

Spatial filtering of the generated pattern defines the output ports of the device, and measurements of the switch response show that it can be actuated by as few as 600~$\pm 40$~photons. For a switch beam with 1/e field radius $w_0=185\,\pm5\,\mu$m, the corresponding energy density is $5.4\,\pm0.7\times10^{-4}$ photons/\lambdasquared, or about a factor of five times greater than the best electromagnetically-induced transparency (EIT) switch reported to date \cite{Zhang_2007aa}.

The response time of the switch depends on the strength of the perturbation and increases for low switch beam power levels. Figure~\ref{fig:intro_photon_number} shows the behavior of the switch response time as a function of switch beam power. Also shown is the number of photons required to actuate the switch. I observe switch response with as few as 600~$\pm 40$ input photons. This device exhibits sensitivity that is comparable to the best reported EIT-based switch to date, however, single-photon switching is not possible with my system.  As I discuss in Chapter~\ref{cha:switch}, single photon switching may be achieved by extending this concept to related systems such as an anisotropic samples of cold atoms. The ultimate performance limitations of this device are also discussed in Chapter~\ref{cha:toymodel}.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/03_05_2008_1_55_response_times.pdf}
  \end{center}
  \caption[Switch response time and photon number]{Based on the switch beam power, and the response time, I calculate the number of photons required to actuate the switch. The number of switching photons is plotted as a function of input power. The dashed line indicates the fit: $N_p =  7081 P_s + 404$ for $P_s$ in nW.}
  \label{fig:intro_photon_number}
\end{figure}

In Chapter~\ref{cha:nummodel}, I describe a numerical model of a counterpropagating beam system that exhibits pattern formation and sensitive switching. Previous simulations, based on the Kerr model described by Firth and Par\'e, have been conducted by Chang \etal \cite{Chang_1992aa} and exhibit the formation of hexagon patterns for a wide range of simulation parameters. In Chapter~\ref{cha:nummodel}, I extend this result by showing that, not only are hexagonal patterns successfully simulated by this model, the model also describes pattern rotation induced by a very weak external switch beam. Also, I show that the simulated switch responds to switch beams that are six orders of magnitude weaker than the pump beams. Furthermore, this model reproduces qualitatively the relationship between the switch response time and the switch-beam power. Figure~\ref{fig:intro_sim_response_time} shows results from the simulation of all-optical switching with transverse patterns generated by gaussian beams counterpropagating through a Kerr nonlinear medium. The switch response time increases for decreasing switch-beam powers and is qualitatively similar to my experimental observations.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/sim_response_time.pdf}
  \end{center}
  \caption[Simulation of the switch response time.]{Simulation of the switch exhibits an increase in response time for decreasing power that is qualitatively similar to experimental observations. To facilitate comparison to Fig.~\ref{fig:photon_number}(a), the horizontal axis has high switch-beam power to the left and low switch-beam power to the right.}
  \label{fig:intro_sim_response_time}
\end{figure}

The qualitative agreement between this model and my experimental observations indicates that the nonlinear Kerr medium, although different from rubidium vapor in important ways, exhibits many of the features required to describe transverse optical patterns and pattern-based all-optical switching.

In Chapter~\ref{cha:toymodel}, I develop a simple 1D model describing the pattern orientation angle as a function of an applied perturbation. This model is based on arguments of symmetry-breaking and a potential which describes the preferred state of orientation. The potential is motivated solely by the observed patterns generated by my system and does not correspond to any physical property of the system as far as I know. Figure~\ref{fig:toymodel} shows the potential with an applied perturbation which causes two wells to be global minima, leading to a two spot pattern. The potential has the form of six wells evenly spaced around a ring. The relative depth of these wells and of the entire ring correspond to the preference of the system to emit light in a hexagonal pattern. To describe the pattern orientation using this potential, I consider motion only around the ring, and map this motion to a one-dimensional flow around a circle. With this simple model, I show how the preferred orientation changes by perturbing the potential. The response time of the orientation is large for weak perturbations and small for strong perturbations, in agreement with my experimental and simulated observations.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=0.5,viewport=30 80 330 245]{Figures/toy_3d_potential_switch.png}
  \end{center}
  \caption[The potential surface for a simple model that exhibits critical slowing down.]{The potential surface for a simple model that exhibits critical slowing down of the pattern rotation. The pattern orientation corresponding to global minima of this hexagonal potential shows that critical slowing down can account for the increased response time I observe experimentally for weak switch-beam power.}
  \label{fig:toymodel}
\end{figure}

In Chapter~\ref{cha:conclusion}, the final chapter, I review my experimental and numerical results, and discuss possible directions for future work.

My preliminary work, conducted in 2004 and 2005, resulted in an all-optical switch that was the first to demonstrate sensitive all-optical switching with transverse patterns \cite{Dawes_2005aa}. The results of my first switch design were substantially better than previous all-optical switches. I achieved switching with as few as 2,700 photons, where the best competing switches required over 1 million photons. Through the course of my research, many improvements have been made to the design of the all-optical switch described here. These improvements are discussed in detail in Appendix~\ref{cha:early_setup}.
% section overview_of_this_thesis (end)


